Based on the objective function of MSER, the gradient of
\(\tilde{\mathcal {P}}_e(b_{i, k})\)  with respect to (w.r.t.) the hybrid
beamforming matrices is given by \(\nabla \tilde{\mathcal {P}}_e(b_{i,k}) \triangleq \nabla \mathcal {P}_e^R(b_{i, k}) + \nabla \mathcal {P}_e^I(b_{i, k})\) , where \(\nabla \mathcal {P}_e^R(b_{i, k})\) 
and \(\nabla \mathcal {P}_e^I(b_{i, k})\)  denote the gradients w.r.t. the
real part and the imaginary part, respectively. Let us first focus on
deriving the gradients w.r.t. the real parts of \(\lbrace \mathbf {P}_k,\mathbf {W}_{k}, \mathbf {U}_k, \mathbf {F} \rbrace \) . Specifically, computing
the gradient of (REF ), as in [1]}, we
obtain,
